The tasks and responsibilities of a game designer vary dramatically and are affected by the team structure, project type, and the development phase.
To shed some light on what you might be required to do, let's look at a real-world example. Here is a list of tasks I undertook when working with a mid-sized team at a London studio, Space Ape games, on a mobile game, Transformers: Earth Wars. The game could be classified as a part of the Build & Battle genre—similar to Clash of Clans, but with a multitude of unique characters, all with their own attacks and abilities.
In the two and a half year period (one year to soft launch, half a year to polish the game, and a year of live operations), my tasks included: ...